Light-tree routing under optical power budget constraints

Transcription

1 Light-tree routing under optical power budget constraints Yufeng Xin MCNC-RDI, RTP, 3021 Cornwallis Road, Research Triangle Park, North Carolina George N Rouskas Department of Computer Science, North Carolina State University, Raleigh, North Carolina rouskas@cscncsuedu RECEIVED 27 OCTOBER 2003; REVISED 15 MARCH 2004; ACCEPTED 16 MARCH 2004; PUBLISHED 14 APRIL 2004 It has been widely recognized that physical-layer impairments, including power losses, must be taken into account when optical connections are routed in transparent networks We study the problem of constructing light-trees under optical-layer power budget constraints, with a focus on algorithms that can guarantee a certain level of quality for the signals received by the destination nodes We define a new constrained light-tree routing problem by introducing a set of constraints on the source destination paths to account for the power losses at the optical layer We investigate a number of variants of this problem, we characterize their complexity, and we develop a suite of corresponding routing algorithms; one of the algorithms is appropriate for networks with sparse light splitting and/or limited splitting fanout We find that, to guarantee an adequate signal quality and to scale to large destination sets, light-trees must be as balanced as possible Numerical results demonstrate that existing algorithms tend to construct highly unbalanced trees and are thus expected to perform poorly in an optical network setting Our algorithms, on the other hand, are designed to construct balanced trees that, in addition to having good performance in terms of signal quality, also ensure a certain degree of fairness among destination nodes Although we consider only power loss here, the algorithms that we develop could be appropriately modified to account for other physical-layer impairments, such as dispersion 2004 Optical Society of America OCIS codes: , Introduction Over the last few years we have witnessed a wide deployment of point-to-point wavelengthdivision multiplexing (WDM) transmission systems in the Internet infrastructure The rapid advancement and evolution of optical technologies makes it possible to move beyond pointto-point WDM transmission systems to an all-optical backbone network that can take full advantage of the available bandwidth because we eliminate the need for per-hop packet forwarding Such a network will consist of a number of optical cross connects (OXCs) arranged in some arbitrary topology, and its main function will be to provide interconnection to a number of client networks, eg, IP subnetworks running multiprotocol label switching [1] (MPLS) Each OXC can switch the optical signal coming in on a wavelength of an input JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 282

2 fiber link to the same wavelength in an output fiber link The OXC may also be equipped with converters that permit it to switch the optical signal on an incoming wavelength of an input fiber to some other wavelength on an output fiber link The main mechanism of transport in such a network is the lightpath, an optical communication channel established over the network of OXCs that may span a number of fiber links (physical hops) If no wavelength converters are used, a lightpath is associated with the same wavelength on each hop This is the well-known wavelength continuity constraint With converters, a different wavelength on each hop may be used to create a lightpath Thus a lightpath is an end-to-end optical connection established between two subnetworks attached to the optical backbone The concept of a lightpath can be generalized into that of a light-tree [2] which, like a lightpath, is a clear channel originating at a given source node and implemented with a single wavelength But unlike a lightpath, a light-tree has multiple destination nodes, and hence it is a point-to-multipoint channel The physical links implementing a light-tree form a tree, rooted at the source node, rather than a path in the physical topology hence the name Light-trees may be implemented with optical devices known as power splitters at the OXCs A power splitter has the ability to split an incoming signal, arriving at some wavelength λ, into as many as m outgoing signals, m 2; m is referred to as the fanout of the power splitter Each of these m signals is then independently switched to a different output port of the OXC Because of the splitting operation and associated losses, the optical signals resulting from the splitting of the original incoming signal must be amplified before leaving the OXC Also, to ensure the quality of each outgoing signal, the maximum fanout m of the power splitter may have to be limited to a small integer If the OXC is also capable of wavelength conversion, each of the m outgoing signals may be shifted, independently of the others, to a wavelength different than the incoming wavelength λ Otherwise, all m outgoing signals will be on the same wavelength λ Note that, just like with wavelength converter devices, incorporating power splitters within an OXC is expected to increase the network cost because of the large amount of power amplification and the difficulty of fabrication Light-trees have several applications [3] in optical networks, including Optical multicast An attractive feature of light-trees is the inherent capability for performing multicast in the optical domain (as opposed to performing multicast at a higher layer, eg, the network layer, which requires electro-optic conversion) Therefore, light-trees can be useful for transporting high-bandwidth, real-time applications such as high-definition TV (HDTV) We note that TV signals are currently carried over distribution networks having a tree-like physical topology; creating a logical tree topology (light-tree) over an arbitrary physical topology for the distribution of similar applications would be a natural next step Because of the multicast property, we will refer to OXCs equipped with power splitters as multicast-capable OXCs (MC-OXCs) Enhanced virtual connectivity In opaque networks, the virtual degree of connectivity of each node is not tied to the number of its interfaces: electronic routing creates the illusion that a node can reach any other node in the network In transparent networks, on the other hand, the degree of connectivity of each client node (eg, IP/MPLS router) connected to the optical core is limited by its physical degree, ie, the number of its optical transceivers A light-tree service would enable a client node to reach a large number of other client nodes independently of its physical degree, significantly enhancing the virtual connectivity of the network Traffic grooming Generalized MPLS (GMPLS) [4] makes it possible to tunnel a set of MPLS label-switched paths (LSPs) over a wavelength channel Since switching at JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 283

3 OXCs takes place at the granularity of a whole wavelength, a point-to-point lightpath allows the sharing of the wavelength bandwidth only between clients attached to the same ingress and egress OXCs The light-tree concept offers a way to overcome this constraint, since it allows for the grooming and tunneling of a number of lower rate point-to-point LSPs to several destinations, regardless of the egress OXC to which these destinations attach In this paper we study the problem of light-tree routing in optical networks with lightsplitting capabilities Since the effects of light splitting and power attenuation on optical signals are only partially mitigated by amplification, our focus is on algorithms that can guarantee a certain level of quality for the signals received by the destination nodes The problem of signal quality does not arise in the context of multicast above the optical layer, and, to the best of our knowledge, it has not been directly addressed in the literature We define a new constrained light-tree routing problem by introducing a set of constraints on the source destination paths to account for the power losses at the optical layer We investigate a number of variants of this problem, and we prove that they are all NP-complete We also develop a suite of corresponding routing algorithms, one of which can be applied to networks with sparse light splitting and/or limited splitting fanout One significant result of our study is that, in order to guarantee an adequate signal quality and to scale to large destination sets, light-trees must be balanced, or distance-weighted balanced (a term we define below) Numerical results demonstrate that existing algorithms tend to construct highly unbalanced trees and are thus expected to perform poorly in an optical network setting Our algorithms, on the other hand, are designed to construct balanced trees that, in addition to having good performance in terms of signal quality, also ensure a certain degree of fairness among destination nodes We emphasize that we consider just the light-tree routing problem in this study, and we do not address wavelength assignment Also, our formulation does not account for signal degradation due to potential interference between light-trees assigned wavelengths that are close to each other Both these problems are important and will be the subject of future research The paper is organized as follows In the remainder of this section we review related work on multicast routing, both in general and in the context of WDM networks In Section 2, we describe the multicast optical network under study, and we develop a model to account for optical signal losses in the network In Section 3 we introduce the problem of constructing light-trees under constraints that ensure the quality of the optical signals received at destination nodes We define three versions of the light-tree routing problem in Section 4, mainly differing on which type of power loss is the dominant factor for signal degradation We characterize the complexity of all three versions of the problem, and we provide light-tree routing algorithms for each We present numerical results in Section 5, and we conclude the paper in Section 6 1A 1A1 Related Research The Steiner Tree Problem To make efficient use of bandwidth in point-to-point networks, the typical approach for multicast communication is to build a multicast tree rooted at the source and spanning all the destinations in a given multicast group Usually, a cost is assigned to each link of the network, and the objective is to determine the tree of minimum cost This is the famous Steiner tree problem [5] in graph theory, which is known to be NP-complete [6] when the multicast group has more than two members Several heuristics and approximation schemes have been developed for the Steiner tree problem These algorithms can be categorized roughly into the following three groups: JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 284

4 Shortest-path-based heuristics (SPH) This algorithm [7] initializes the Steiner tree to the shortest path from the source to an arbitrary multicast member It then repeatedly includes a new member by adding the shortest path between this member to the current partial tree, until all members have joined the tree Many variants of this algorithm have been developed to improve the quality of the final tree, such as including the members in the order determined by their distance to the multicast tree [8] instead of random inclusion, or growing the Steiner tree from the destinations [9] instead of from the source Spanning tree-based heuristics (STH) This algorithm [10] first constructs a closure graph of the multicast nodes from the original graph, using the cost of the shortest path between each pair of members A minimum spanning tree of the closure graph is obtained (in polynomial time), and then the shortest paths in the original graph are used to replace the edges of this minimum spanning tree Finally, the multicast tree is obtained by removing any cycles This approach yields an approximation algorithm with a ratio of 2 Metaheuristics Metaheuristics such as simulation annealing [11], genetic algorithms [12], and Tabu search [13] have been investigated to solve the Steiner tree problem and have been shown to perform well on average In practice, the nature of some multicast applications is such that the routing tree must satisfy certain constraints related to physical limitations (eg, a limited fanout capability) or the desired quality of service (eg, an upper bound on the end-to-end delay along any path of the tree) Constrained Steiner tree problems are at least as hard as the unconstrained one, and for certain constraints it has been shown that no polynomial-time approximation scheme exists Several heuristics have been developed to compute constrained trees, most of which are based on the above heuristics for the unconstrained Steiner tree problem For instance, the KPP algorithm [14] uses an approach similar to STH to compute an approximate Steiner tree in which the end-to-end delay along any path from the source to a destination node is bounded The degree-constrained Steiner tree problem [15, 16], in which it is assumed that some nodes may not support multicast (ie, they cannot be used as branching points) or have a limited fanout capability has also been studied and appropriate heuristics have been proposed A constrained multicast tree problem in which the objective is to bound both the end-to-end delay and the delay variation among all source destination paths has also been studied [17]; the total cost of the tree was not considered in that paper, but it was shown that constructing such a constrained tree is an NP-complete problem 1A2 Light-Tree Routing With recent advances in MC-OXC technology [18], it is now possible to envision a future backbone network environment that provides a practical multicast service at the optical layer Such a service will rely on GMPLS-related protocols to establish light-trees on demand [3] Our research, as well as previous studies of light-trees that we review in this section, makes the assumption that optical networks of OXCs can be practically extended to implement a multicast service at the optical layer when the OXC functionality is augmented to provide additional capabilities in terms of (1) hardware (ie, power splitter and delivery switches, as we describe in Section 2) to split an incoming signal into multiple outgoing signals, and independently switch the new signals to different output ports; and (2) software (ie, GMPLS-related protocols) for dynamically signalling the creation of the light-tree and establishing the multicast connection Although the problem of establishing a light-tree that spans a given source and a set of destination nodes bears some similarities to the Steiner tree problem, the nature of optical JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 285

5 multicast introduces several new issues and complexities Splitting an optical signal introduces losses, a problem not encountered in electronic packet- or circuit-switched networks, and thus, not addressed by existing routing tree algorithms Even in the presence of optical amplifiers, this signal loss imposes a hard upper bound on the number of times a signal can be split, as well as on the number of hops that the signal can travel after every split operation In the absence of wavelength conversion in the network (or even in networks with limited or sparse conversion capability), multicast routing is tightly coupled to wavelength allocation, an issue that does not arise in electronic networks Also, optical networks may have only a sparse multicast switching capability; ie, only a subset of the OXCs may be multicast capable When only a few MC-OXCs are present in the network, a feasible multicast tree may not exist, and therefore the heuristics for degree-constrained multicast [15] are not applicable at all Finally, the problems of capacity planning of MC-OXCs and multicast routing strongly depend on each other Several recent research efforts have aimed to address some of the problems associated with optical multicast and light-tree establishment, including studies of wavelength assignment in the presence of multicast [19 21] and multicast routing algorithms for networks with a sparse light splitting capability [22 24] To deal with the fact that a feasible multicast tree may not exist for a given source and destination set, the concept of a light-forest has been proposed [24] In general, all the multicast routing algorithms for optical networks assume unlimited fanout capacity at MC-OXCs, and each tree of a given light-forest must be assigned a different wavelength The problem of optimally placing a small number of MC-OXCs in a WDM network has been studied in Ref [25] Finally, two designs for MC-OXCs have been proposed The first is based on the splitter-and-delivery architecture [18], whereas the second is an enhancement of the former that results in better power efficiency [26] The reader is also referred to a recent comprehensive survey [3] of the optical multicast problem by one of the authors 2 The Multicast Optical Network We consider an optical WDM network with N nodes interconnected by fiber links Each of the links is capable of carrying W wavelengths, and each of the nodes is equipped with an OXC with P input ports and P output ports The OXC at (some of) the nodes is multicastcapable (MC-OXC) A P P MC-OXC consists of a set of W P P splitter-and-delivery (SaD) switches, one for each wavelength; Fig 1 shows a 3 3 MC-OXC for W = 2 wavelengths In addition to the W SaD switches, P demultiplexers (respectively, multiplexers) are used to extract (respectively, combine) individual wavelengths The SaD switch design was first proposed [18] and was later modified [26] in order to reduce cost and improve power efficiency A P P SaD switch, as originally proposed [18], is shown in Fig 2 It consists of P power splitters, P 2 optical gates (to reduce the excessive cross talk), and P photonic switches We assume that the splitters are configurable, in that they can be instructed to split the incoming signal into m output signals, m = 1,,P; note that m = 1 corresponds to no power splitting, ie, no multicast, whereas m = P corresponds to a broadcast operation By appropriate configuration of the corresponding m 2 1 photonic switches, each of the m signals resulting from the splitting operation can be switched to the desired output ports In a transparent network, optical signals experience losses as they travel from source to destination node We distinguish two types of losses: 1 Signal attenuation This is due to the propagation of light along the fibers between the source and destination nodes Optical amplifiers (erbium-doped fiber amplifiers, ED- FAs) are used along the optical paths to boost the power of the information-carrying signals in order to compensate for the signal attenuation However, optical power JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 286

6 Input port 1 Input port 2 Input port 3 λ 2 λ 1 λ 2 λ 1 λ 1 λ 1 λ 2 SaD Switch SaD Switch λ 1 λ 1 λ 2 λ 2 λ 2 Output port 1 Output port 2 Output port 3 Fibers Wavelength Demux Wavelength Mux Fig 1 A 3 3 MC-OXC based on the SaD switch architecture, W = 2 2A amplification is not perfect, and there is a limit on the number of times a signal may be amplified Thus, it has been suggested [27] that power attenuation (along with other physical layer impairments, such as dispersion) be taken into account when lightpaths are routed in a transparent optical network 2 Splitting loss An m-way splitter (similar to those shown in Fig 2) is an optical device that splits an input signal among m outputs For an ideal device, the power of each output is (1/m)th of that of the original signal; in practice, the splitting operation introduces additional losses and the power of each output is lower than that of the ideal case Splitting losses occur within MC-OXCs at the branch points of light-trees carrying point-to-multipoint signals Although amplification may partially compensate for the power loss due to light splitting, it is clear that this type of loss must be taken into account for light-tree routing The next two subsections discuss the two types of losses in more detail Power Attenuation Along a Fiber Link The output power P out at the end of a fiber of length L is related to the input power P in by P out = P in e αl, (1) where α is the fiber attenuation ratio [28]; near 1550 nm, we have that 434α = 02 = α db In general, distributed-feedback (DFB) lasers put out 50 mw (17 db) of power after the output signal is boosted by an amplifier, whereas the sensitivity of avalanche photodiode (APD) receivers at 25 Gbit/s is 34 db [29] Therefore, from Eq (1), we obtain the maximum transmission distance in a fiber as L max = 255 km For any fiber link whose length is greater than L max, a number of EDFA amplifiers must be added to compensate for the power attenuation so that the receiving power at the end of the fiber is no less than 34 db When optical amplifiers are used, other constraints must be considered, including the maximum permissible power on a fiber, the effects of fiber nonlinearities, and the receiver sensitivity Consequently, in current practice, amplifier spacings range from 20 to 100 km Assume that the span of length between two consecutive amplifiers (EDFAs) in the optical network is S amp and that the gain of each amplifier is denoted by G amp (S amp and G amp are assumed to be parameters that are fixed for a particular fiber system) Then, the power received at the end of a fiber link of length L is related to the input power as follows: ( P out = P in Gamp e αs ) ( L/S amp amp = P in G 1/S amp amp e α) L = Pin Q L, (2) JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 287

7 Inputs 1 2 P Outputs 1 2 P Splitter Gate 2x1 Switch Fig 2 P P SaD switch where Q = G 1/S amp amp e α < 1 is a constant determined by the fiber system Expression (2) describes the signal attenuation within a fiber link equipped with optical amplifiers, as a function of the link length L 2B Power Loss Due to Light Splitting at the MC-OXC Let us now consider a signal that arrives at some input port of an MC-OXC such as the one shown in Fig 1 This signal is split into m output signals at the SaD switch corresponding to the input s wavelength The m output signals are then switched to the appropriate output ports of the MC-OXC We assume that the power splitters at the SaD switch (refer to Fig 2) are configurable, such that a multicast optical signal does not always need to be split P times, where P is the number of input output ports of the SaD switch Instead, the multicast signal is split into exactly m signals, m = 1,,P, where m is the out-degree of the node in the corresponding light-tree Configurability is made possible by new devices such as the compact multimode interference couplers with tunable power splitting ratios that were reported recently [30] We also assume that the tunable power splitting ratio can be controlled by the multicast signaling protocol, making it possible to realize MC-OXCs with any desirable fanout m,m = 1,,P Given these assumptions, the power loss (in decibels) at an MC-OXC for an input signal that is split into m output signals is given by [26]: Loss SaD = 10 log 10 m+β(p) (3) In the above expression, the term β(p) captures losses due to the multiplexing and demultiplexing of signals, as well as the insertion and coupling losses at the 2 1 switching elements (refer to Fig 2) Since the number of switching elements in the signal path is equal to the number P of input output ports of the SaD switch, then this term is a function of P From Eq (3), we can now derive the output power of each of the m output signals as a JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 288

8 function of the input power as follows: P out = β(p) m P in P in m (4) Expression (4) assumes that signals are not amplified as they leave the MC-OXC To compensate for the power loss due to light splitting, optical amplifiers may be placed at the output ports of the MC-OXC Let G amp denote the gain of an amplifier, and define R = 10 β(p) 10 G amp R is a constant for a given SaD switch, and is determined by the number of ports P of the switch, the losses incurred at the various elements of the switch, and the amplifier gain Then, the output power of a signal that has undergone m-way splitting is given by P out = RP in m P in (5) 3 The Light-Tree Routing Problem We represent a network of MC-OXCs by a simple graph G = (V,A) V denotes the set of nodes (ie, MC-OXCs), and A, the set of arcs, corresponds to the set of (unidirectional) fiber links connecting the nodes We will also use N = V to refer to the number of nodes in the network We define a distance functiond : A R +, which assigns a nonnegative weight to each fiber link in the network More specifically, the value D (l) associated with link l = (u,v) A,u,v V, is the geographical distance that the optical signal travels along the link l from node u to node v Under the light-tree routing scenario that we are considering, an optical signal originating at some source node s V in the network must be delivered to a set M V {s} of destination nodes In general, several point-to-multipoint sessions may proceed concurrently within the network, each characterized by a source node and a destination set We assume that communication in the network is connection-oriented, and that point-tomultipoint connections are established by issuing a connect request; similarly, at the conclusion of a session a disconnect request is issued In response to a connect request, and prior to any optical signal been transmitted from the source to the destinations, a connection establishment process is initiated Central to the connection establishment is the determination of a light-tree, ie, a set of paths between the source and the destinations, over which the optical signal will be carried for the duration of the point-to-multipoint session Let s and M be the source and destination set, respectively, of a certain point-tomultipoint session We let T = (V T,A T ) denote the light-tree, rooted at s, for this session The light-tree is a subgraph of G (ie, V T V and A T A) spanning s and the nodes in M (that is, M {s} V T ) In addition, V T may contain relay nodes, that is, nodes intermediate to the path from the source to a destination Relay nodes do not terminate the optical signal transmitted by the source node s; rather, they simply split and/or switch the signal toward the downstream links of the light-tree We let H T (s,v) denote the unique path from source s to destination v M in the light-tree T We define P in (s) as the power of the optical signal injected into the network by the source node s, and P out (s,v) as the power of the optical signal received by destination v M The output power P out (s,v) at destination v is related to the input power at the source s through the following expression: P out (s,v) = P in (s) L (atten) (s,v) L (split) (s,v) (6) In the above expression, parameter L (atten) (s,v) [respectively, L (split) (s,v)] accounts for the power loss due to attenuation (respectively, light splitting) along the path from s to v in the light-tree T ; we assume that both parameters include the effects of amplification JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 289

9 Recall that expression (2) relates the input and output signal power for a single fiber link The expression can be generalized to a path from a source s to a destination v in a straightforward manner, allowing us to express L (atten) (s,v) as follows: L (atten) (s,v) = l H T (s,v) Q D(l) = Q l H T (s,v) D(l) < 1 (7) Similarly, we can obtain an expression for L (split) by considering all MC-OXCs in the path from s to v in the light-tree T, and applying expression (5) Let us define F T (u) as the fanout of the MC-OXC at node u of the light-tree T, with respect to the optical signal carried on this light-tree (note that node v may be part of a different light-tree T, with a different source and destination set; its fanout with respect to T may be different than its fanout with respect to T ) The fanout F T (u) corresponds to the quantity m in expression (5) As a result, we obtain L (split) R (s,v) = < 1 (8) F T (u) u H T (s,v) We note that, as we explained in the previous section, quantities Q and R in expressions (7) and (8) are constants for a given optical network 3A Path Constraints to Ensure Optical Signal Quality We now introduce two parameters that can be used to characterize the quality of the lighttree as perceived by the application making use of the point-to-multipoint optical communication These parameters relate the end-to-end power loss along individual source destination paths to the desired level of signal power at the receivers, as follows Source destination loss tolerance, Parameter represents an upper bound on the acceptable end-to-end power loss along any path from the source to a destination node This parameter reflects the fact that if the optical signal power falls below the receiver sensitivity, then the information carried by the signal cannot be recovered Interdestination loss variation tolerance, δ Parameter δ is the maximum difference between the end-to-end losses along the paths from the source to any two destination nodes that can be tolerated by the application This parameter can be thought of as a measure of fairness among the destination nodes of the light-tree By supplying values for parameters and δ, the application in effect imposes a set of constraints on the optical signal power at the receivers of the light-tree: P out (s,v) P in (s) v M, 1, (9) 1 δ P out(s,v) δ v,u M,δ 1 (10) P out (s,u) We will refer to Eq (9) as the source destination loss constraint, and Eq (10) will be called the interdestination loss variation constraint We will also say that tree T is a feasible lighttree for a point-to-multipoint session with source s and destination set M, if and only if T satisfies both Eq (9) and Eq (10) Note that, in order for the application to proceed, it is necessary and sufficient that a single feasible light-tree be constructed, since any feasible tree can meet the quality of service requirements as expressed by parameters and δ JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 290

10 4 Optical Signal Power Constrained Light-Trees Let and δ be the loss and loss variation tolerances, respectively, as specified by a client application that wishes to initiate a point-to-multipoint session Our objective is to determine a light-tree such that the power losses along all source destination paths in the tree are within the two tolerances This problem, which we will call the power constrained light-tree (PCLT) problem, can be formally expressed as follows Problem 41 (PCLT) Given a network G = (V, A), a source node s V, a destination set M V {s}, a distance function D : A R +, a loss tolerance, and a loss variation tolerance δ, does there exist a light-tree T = (V T,A T ) spanning s and the nodes in M, that satisfies both constraints (9) and (10)? In the next three subsections we study three variants of the PCLT problem The variants mainly differ in the assumptions made regarding the degree to which each of the two types of power loss (ie, loss due to attenuation or light splitting) affects the quality of the received signal As we explain, the assumptions depend on the geographical span of the light-tree and the size of the destination set, and it is possible that different variants of the PCLT problem apply to different light-trees within the same optical network Therefore, we characterize the complexity of, and provide light-tree algorithms for, all three variants of the PCLT problem 4A The PCLT Problem Under Power Attenuation Only Let us first consider the PCLT problem under the assumption that power attenuation is the dominant factor in determining the signal quality at the receivers of the light-tree In other words, we assume that L (split) (s,v) 1 in expression (6), for all destinations v This is a reasonable assumption when (i) the source of the point-to-multipoint session and the destination nodes are separated by large geographical distances, and/or (ii) there is a small number of destination nodes; thus, the optical signal needs to undergo a only small number of splitting operations In this case, we can use Eq (7) to rewrite the source destination constraint (9) and the interdestination loss variation constraint (10) as follows: L (atten) (s,v) Q l H T (s,v) D(l) l H T (s,v) D (l) log Q v M, (11) 1 δ L(atten) (s,v) L (atten) (s,u) δ Q l HT (s,v) D(l) l H T (s,u) D(l) δ D (l) D (l) log Q δ v,u M (12) l H T (s,v) l H T (s,u) Note that the final step in constraint (11) is due to the fact that constants Q and are such that 0 <,Q < 1 An interesting observation regarding constraints (11) and (12) is that they represent two conflicting objectives Indeed, the loss constraint (11) dictates that short paths be used But choosing the shortest paths may lead to a violation of the loss variation constraint (12) among nodes that are close to the source and nodes that are far away from it Consequently, it may be necessary to select longer paths for some nodes in order to satisfy the latter constraint Then, the problem of finding a feasible light-tree becomes one of selecting paths in a way that strikes a balance between these two objectives The PCLT problem with constraints (11) and (12) is equivalent to the delay- and delay variation-bounded multicast tree (DVBMT) problem [17, 31] Specifically, the loss constraint (11) is equivalent to the delay constraint of DVBMT, whereas the loss variation JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 291

11 constraint (12) is equivalent to the delay variation constraint of DVBMT We have proved [17] that the DVBMT problem is NP-complete whenever the size of the destination set M 2 Consequently, if we ignore the power loss resulting from the splitting of the optical signal at the branch nodes of the light-tree, the PCLT problem is also NP-complete In this case, the heuristics developed for DVBMT [17, 31] can be applied directly to construct a light-tree that satisfies both constraints (11) and (12) 4B The PCLT Problem under Splitting Losses Only Let us now turn our attention to the case when signal attenuation is negligible [ie, L (atten) 1 in expression (6)], and power loss due to light splitting is the dominant factor affecting signal quality at the receivers This situation may arise when (i) the destination set includes a large number of nodes, and/or (ii) the source and destination nodes are located in close proximity to each other We can then use expression (8) to rewrite constraints (9) and (10) as follows [recall that F T (w) is the fanout of node w with respect to light-tree T ; in other words, it denotes the number of times the optical signal traveling along light-tree T is split at node w] R v M, (13) F T (w) w H T (s,v) 1 δ w H T (s,v) w HT (s,u) R F T (w) R F T (w) δ u,v M (14) Let us interpret constraints (13) and (14) Without loss of generality, let us assume that R = 1, ie, that the power of each of the F T (w) output signals at node w is [1/F T (w)]th of that of the input signal; our conclusions are valid even when R > 1 When R = 1, the denominator of the left-hand side of expression (13) corresponds to the product w HT (s,v) F T (w) along the path from the source s to destination v We will call this product the split ratio of node v, and its inverse corresponds to the residual power of the optical signal received at node v after all the splits along the path We can see that constraint (13) imposes an upper bound on the split ratio on the path to each destination node in set M Let us now turn our attention to constraint (14) When R = 1, it states that the split ratios of any two paths from the source to two destination nodes v and u should be within a tight range from each other, where the tightness of the range is determined by parameter δ Therefore, this constraint suggests that light-trees must be as balanced as possible To see why, assume that a light-tree is constructed for a set of K destinations such that one destination node, say v, is directly connected to the root (source) while the remaining K 1 nodes are all in a different subtree connected to the root It is clear that, even after amplification (ie, R > 1), node v will receive a signal of better quality than the other K 1 destinations: the signal arriving at node v is of the same quality as the one traveling toward the other subtree, but the latter signal will have to be split several times (and thus, it will degrade further) before it reaches each of the K 1 destinations in the subtree Such an unbalanced tree has two important disadvantages First, it introduces unfairness, since receivers at small depth in the (logical) tree receive a signal of better quality than receivers at large depth, independently of their geographical distance to the source Second, it is not scalable, since it may introduce excessive losses that make it impossible to deliver a signal to a given number of destinations To see this, consider a worst case scenario where the tree is a binary one and is recursively constructed such that the left subtree consists of exactly one receiver, while the right subtree contains all remaining receivers and consists of left and right subtrees in a similar way It is easy to see that the receiver at depth one (in the left subtree of the whole tree) receives a signal that has undergone one split and its power is one-half of that of the original signal On the other hand, the receiver at depth K (the JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 292

12 rightmost ( leaf of the tree) receives a signal that is the result of K splits, and its power is 1/2 K ) th of that of the original signal While extreme, this scenario illustrates the pitfalls of unbalanced trees for the multicast of optical signals The requirement that the light-tree be as balanced as possible is a direct consequence of the fact that when an optical signal undergoes m-way splitting; its power is equally divided among the m output signals Thus, this requirement is unique to optical layer multicast To the best of our knowledge, the problem of constructing balanced multicast trees has not been studied in the literature, since it does not arise in the context of multicast above the optical layer We now prove the problem of constructing balanced multicast trees to be NPcomplete In Subsection 4B1 we present a suite of heuristics to obtain balanced light-trees that satisfy constraints (13) and (14) Our proof is by reduction from the exact cover by three-sets (X3C) problem [32], a well-known NP-complete problem defined as follows: Definition 41 (X3C) Given a set S = {S i } with 3k elements for some natural number k and a collection Y = { Y j } of subsets of the set, each of which contains exactly three elements, do there exist in the collection Y k subsets that together cover the set S? Theorem 41 The PCLT problem under constraints (13) and (14) is NP-complete Proof Clearly, PCLT belongs in the class NP, since a solution to the PCLT problem can be verified in polynomial time We now transform the NP-complete X3C problem to PCLT Consider an arbitrary instance of the X3C problem consisting of (i) a set S = {S i } of elements, where S = 3k for some natural number k, and (ii) a collection Y = { Y j } of subsets of S, each subset containing exactly three elements of S Let m = S,n = Y We construct a corresponding instance of PCLT as follows The graph G = (V,A) has n+m+1 nodes, with V = {s,y 1,Y 2,,Y n,s 1,S 2,,S m }, where s is the source node and M = S = {S i } is the destination set of the light-tree The set A of links is A = {(s,y 1 ),(s,y 2 ),,(s,y n )} { (Y j,s i ) Y j Y S i Y j } (15) In other words, there is a link from s to every node Y i, and a link from every node Y i to every node S j that is a member of Y i (see Fig 3) The distance function is defined as D (l) = 1, l A (in fact, the distance function can be arbitrary; since this variant of PCLT neglects power attenuation, constraints (13) and (14) do not depend on the link weights) Finally, the loss and loss variation tolerances are = 1/3k and δ = 1, respectively It is obvious that this transformation can be performed in polynomial time We now show that a feasible light-tree for the PCLT problem exists if and only if set S has an exact cover If S has a cover X = { Y π1,y π2,,y πk }, the tree containing the source s, the set of nodes X = { Y π1,y π2,,y πk }, and the set of nodes S = {Si } is a feasible solution for PCLT This is because the split ratio of each destination node S i is equal to 1/3k, and the tree satisfies both constraints (13) and (14) Conversely, let T be a feasible light-tree for PCLT Then, T must contain the source node s, all destination nodes S i, and a subset X of Y = { Y j } Since δ = 1, all destination nodes Si have the same split ratio By construction of the PCLT instance, each destination node S i must have exactly one parent in the lighttree T : if some node S i had more than one parents a loop would exist (contradicting the hypothesis that T is a tree), and if it had no parent, it would not be connected to the tree T (again contradicting the hypothesis that T is a solution to the PCLT problem, ie, it spans all destination nodes) Therefore, the nodes in the subset X of Y contained in the light-tree T exactly cover the set S, implying that X is a solution to the instance of the X3C problem JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 293

13 s Y1 Y2 Y3 Y4 Y5 S1 S2 S3 S4 S5 S6 S7 S8 S9 Fig 3 Instance of PCLT corresponding to an instance of X3C with k = 3, S = {S 1,,S 9 }, Y 1 = {S 1,S 2,S 4 }, Y 2 = {S 2,S 4,S 5 }, Y 3 = {S 3,S 5,S 7 }, Y 4 = {S 4,S 6,S 7 }, Y 5 = {S 6,S 8,S 9 }, and exact cover {Y 1,Y 3,Y 5 }; the light-tree T is denoted by dashed lines 4B1 Balanced Light-Tree Algorithm We now present an algorithm for the version of the PCLT problem discussed above The objective of any such algorithm would be to construct a feasible light-tree, ie, one that satisfies both constraints (13) and (14) Note, however, that since the PCLT problem is NPcomplete, any polynomial-time algorithm may fail to construct a feasible light-tree for a given problem instance, even if one exists The algorithm we present can be used to search through the space of candidate trees (ie, trees spanning s and the nodes in M) for a feasible solution to the PCLT problem Our algorithm either returns a feasible tree, or, having failed to discover such a tree, it returns one for which (i) the maximum split ratio of any node in M, and (ii) the maximum difference between the split ratios of any pair of nodes in M, are minimum over all trees considered by the algorithm The balanced light-tree (BLT) algorithm, described in detail in Algorithm 1, takes as input an initial tree T 0 spanning the source s and destination nodes in M; the issue of constructing this initial tree is addressed shortly In general, tree T 0 may be infeasible, ie, it may violate (13) and/or (14) The key part of the BLT algorithm is the tree balancing procedure that is implemented by the while loop in steps 4 18 of Algorithm 1 Consider an intermediate light-tree T, and let u (respectively, v) denote the leaf node with maximum (respectively, minimum) split ratio The idea behind the BLT algorithm is to delete node u from T, and add it back to the tree by connecting it to some node y in the path from source s to v Doing so reduces the split ratio of node u, but it also increases the split ratio of all nodes below node y in the tree; therefore, this pair of delete add operations is performed only if it does not increase the split ratio of any node beyond that of node u (refer also to the if statement in steps of Algorithm 1) Thus, after each iteration of the algorithm, the split ratio of the node with the maximum value is decreased, in an attempt to satisfy constraint (13) While the split ratio of some other node(s) is increased, it does not increase beyond the previous maximum value As a result, the difference between the maximum and minimum split ratio values also decreases with each iteration, as required by constraint (14) The algorithm terminates after a certain number of iterations, or if two successive iterations fail to reduce the maximum split ratio; the latter condition is not shown in Algorithm 1 in order to keep the pseudocode description simple To completely specify the BLT algorithm, we now explain how to select the node y in the path from s to v (the node with the minimum split ratio) to connect node u (the one with the maximum split ratio) Let Y denote the number of nodes in the path from source s to node v We consider three different criteria for selecting a node y Y to which to connect JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 294

14 Algorithm 1 General Balanced Light-Tree (BLT) Algorithm Input: A graph G = (V,A) representing the network of MC-OXCs, a source node s V, a destination set M V, a loss tolerance, a loss variation tolerance δ, and an initial lighttree T 0 spanning the set {s M} Output: A light-tree T f spanning the set {s M}, and such that either (i) T f is feasible, or (ii) the difference between the maximum and minimum split ratio of any two nodes in M is minimum 1 begin 2 T T 0 // Initialize the light-tree 3 h 1 // Number of iterations 4 while (h I) //I is the maximum number of iterations 5 Use depth-first search to calculate the split ratio of all nodes in M 6 if (light-tree T is feasible) then return T 7 u leaf node with maximum split ratio 8 v leaf node with minimum split ratio 9 w the first node in the path from u to s in T such that w M or w has a fanout > 1 10 Y set of nodes in the path from v to s in T 12 In G, compute shortest paths from u to every node in Y 13 y a node in Y selected based on one of the criteria in Section?? 14 if (max split ratio of T does not increase) then 15 Delete the path from w to u in tree T // Delete the node with the maximum split ratio 16 Add the shortest path from y to u to T // Add the node back to T on a different path 17 end if 18 end while 19 return T 20 end algorithm node u, resulting in three variants of the BLT algorithm 1 Shortest path (BLT-SP) In this variant, we select node y such that the path from y to u is shortest among the paths from any node in Y to u 2 Minimum split ratio (BLT-MSR) In this case, node y is one with the smallest split ratio among all nodes in Y 3 Degree constraint (BLT-D) This is similar to BLT-MSR, except that the node y selected must be such that its fanout is no more than a maximum value F F may correspond to the maximum fanout capacity of the SaD switches at each MC-OXC With this selection criterion, the resulting light-tree will have a bounded degree (fanout) Note that, if we use a different value of F for each node in the network, then the algorithm can be used in optical networks with sparse light splitting, since multicastincapable OXCs can be accounted for by letting F = 1 for these nodes Finally, we use the SPH algorithm [8] to construct an initial tree T 0 that spans the source node s and the destination set M The SPH algorithm is a fast algorithm that has been used successfully as a starting point for several constrained Steiner tree problems [15] The algorithm starts with a partial tree consisting of the shortest path from the source s to some JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 295

15 destination node It then repeatedly extends the partial tree to another destination node u, until all destination nodes have been included A new destination node u is connected to the partial tree by including the shortest path from some node y of the tree to u Therefore, the issue arises of selecting the node y of the partial tree to which to connect node u For each variant of the BLT algorithm, we use the corresponding selection criterion to select node y of the partial tree Regarding the complexity of the BLT algorithm shown in Algorithm 1, it is straightforward to verify that the worst-case running time is O ( N 2 I ), where N is the number of nodes in the network and I is the maximum number of iterations for the while loop in steps 4 18 Note that, at the end of each iteration of the while loop, the maximum split ratio is decreased by at least one, and the minimum split ratio increases by at least one A lower bound on the minimum split ratio is the number M of destinations, since the signal has to be split at least M times In our experiments (refer to Section 5), we have found that, typically, the initial value of the maximum split ratio is around N/2, where N is the number of network nodes (OXCs) Therefore, in the worst case, the number I of iterations is O(N/2 M) Finally, we note that the worst-case complexity is the same for all three variants of the BLT algorithm 4C The General PCLT Problem We now consider the most general version of the PCLT, which arises when both signal attenuation and light splitting contribute to the degradation of the quality of the signal as it travels through the optical network In this case, the signal power received at each destination node is related to the signal power emitted by the source node through expressions (6), (7), (8), and the light-tree must be constructed such that constraints (9) and (10) be satisfied Clearly, this version of the PCLT problem is also NP-complete, since it includes as special cases the two versions studied in Subsections 4A and 4B, both of which are NP-complete An interesting observation regarding this general version of the PCLT problem is that there is a trade-off between the number of times a signal may be split and the distance that the signal can travel Signals that have been split multiple times may not be able to travel over large distances, even after amplification, and vice versa This trade-off, which is unique to optical networks, is not taken into account by existing multicast routing algorithms In this case, it would be desirable to have receivers which are far away (in terms of distance traveled by the optical signal) from the source, be closer to the source in the (logical) light-tree This way, the signal arriving to these receivers will have undergone a smaller number of splits In this case, the resulting light-tree will not necessarily be balanced (in the traditional definition of the term), but rather it must be balanced in a manner that accounts for the geographical locations of the various receivers relative to the source In other words, the number of signal splits for each receiver must be appropriately weighted by the distance to the receiver From the above observations, we modify the BLT algorithm shown in Algorithm 1 to construct distance-weighted balanced light-trees; we will call this algorithm weighted BLT (WBLT) The main idea is to consider the tree node with the largest total loss and attempt to reduce its splitting loss by moving it closer to the source in the logical light-tree Doing so may increase the attenuation loss (since the node may be added to the tree on a longer path), but it will also decrease its splitting loss, possibly resulting in a smaller total loss This weighted balancing procedure can be accomplished by making the following small changes in the algorithm of Algorithm 1: in step 7 (respectively, step 8), select the node with the maximum (respectively, minimum) total loss, and in the if statement in step 14, check whether the maximum total loss at any node of the tree increases Otherwise, the algorithm remains unchanged Note that, since there are three variants of the BLT algorithm, we also JON 3213 May 2004 / Vol 3, No 5 / JOURNAL OF OPTICAL NETWORKING 296